A higher-order asymptotic expansion of the Krawtchouk polynomials

The paper extends the classical result on the convergence of Krawtchouk polynomials to Hermite polynomials. We provide a uniform asymptotic expansion of Krawtchouk polynomials in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation.